1. How many different non-congruent isosceles triangles can be formed by connecting three of the dots in a $4\times4$ square array of dots like the one shown below? [asy] size(50); dot((0,0));dot((0,1));dot((0,2));dot((0,3)); dot((1,0));dot((1,1));dot((1,2));dot((1,3)); dot((2,0));dot((2,1));dot((2,2));dot((2,3)); dot((3,0));dot((3,1));dot((3,2));dot((3,3)); [/asy]

Two triangles are congruent if they have the same traced outline, possibly up to rotating and flipping. This is equivalent to having the same set of 3 side lengths.

2. I have $5$ different pullover shirts and $4$ different button-down shirts. In how many ways can I choose shirts for the next $9$ days if I insist on wearing pullover shirts two days in a row at least once? Assume that I wear one shirt each day, and every shirt gets worn once.

Guest Jun 3, 2018